Coupling a membrane with pretension to point displacement Michael Alletto

• contributor: Michael Alletto
• affiliation: WRD MS
• contact: click here for email address
• OpenFOAM versions: v2112
• Published under: CC BY-NC-ND license (creative commons licenses)

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You can download the case https://gitlab.com/mAlletto/openfoamtutorials/-/tree/master/membranBCSend] here.

Introduction

In this small tutorial we will see how to couple two different physical problems in OpenFoam. In this specific case we will use the equation of motion of a membrane with pretension to impose the movement of the mesh at a given boundary. For membranes (which are very thin) the variation of the stress over the thickness is assumed constant. For this reason the governing equation reduce to a two dimensional problem. The equation of motion are described also in Wagner et al [3]. Part of the code presented later is taken from the work mentioned before. If a problem is purely two dimensional, OpenFoam offers the finite area method where a framework to discretize and solve 2D equation is offered. Summing up, we will see how to write advanced boundary condition for the 3D fields present in OpenFoam. In our specific case we will see how to derive a 2D differential equation for the membrane displacement w(x,y), solve it and use the solution to move the mesh at the boundary where our membrane boundary condition is applied. In order to deform the membrane, the pressure of the fluid is used.

Equation of motion

As first point in this tutorial we will briefly derive the equation of motion of a membrane subject to a prestress σ. Note that the derivation of the differential equation for a rod subject to an external force in z-direction and prestretched by a force S (the corresponding one dimensional case of a membrane subject to a prestess σ) can be found in Timoshenko [2] on page 352. Here we will see how to derive the equation for the two dimensional case. The next figure shows a sketch of the membrane subject to a prestress σ and an external pressure p.

The membrane has a thickness h. In order to derive the equation of motion one has to apply Newtons second law to an infinitesimal element with an extension dx and dy in a Cartesian coordinate system:

References

[2] Stephen Timoshenko, Sergius Woinowsky-Krieger, et al. Theory of plates and shells, volume 2. McGraw-hill New York, 1959.

[3] Simon Wagner, Manuel Münsch, and Antonio Delgado. An integrated openfoam membrane fluid-structure interaction solver. OpenFOAM R Journal, 2:48–61, 2022.